In any case, a thinking person will recognise that both parts
of education are important. A more skills oriented person will
recognise that acquiring knowledge and docketing it in a way that
allows us to utilitise it later is an important skill. On the
other hand, one who pursues knowledge will concede that a good
part of our knowledge is the result of attempting to codify
learned skills.

However, like any other "versus" discussion, the pendulum
periodically swings one way or another. In my opinion, currently,
students in India (and those who fund them, i.e. their guardians
and parents) are looking for skills-based education. Rightly or
wrongly, they believe that this is what will put them ahead in
the job market. In my opinion, that is why a large number of
students wish to enrol in "engineering colleges" instead of
"universities". The latter are seen as being primarily the
disseminators of the collective wisdom of the ages, while the
former are seen as places that will teach them the skills that
will be "useful in jobs". It is perhaps thought that with Google,
Wikipedia other source of information, it is no longer necessary
to learn "stuff"; it is more important to learn skills,
especially if one of those skills is that of finding useful stuff
from the store-houses of information.

Students who enroll in engineering colleges and their avatars
are thus very disappointed to find that (in a large majority of
cases) these places are just universities in disguise. This is
doubly so in elite institutes like the IITs since the instructors
there are usually "pursuing knowledge". There is a repeatedly
expressed feeling in diverse student fora that universities are
"factories operated by professors to generate more professors".
Given the paucity of qualified teachers and researchers, this is
not necessarily a poor goal --- even from a job-oriented
point-of-view. However, students who join up for a
technical education donot see
"professor" or "researcher" as one of their career goals!
[1]

Of course, we could attempt to convince/brainwash some
of the students into pursuing these laudable careers, but
that is not likely to succeed in a majority of cases.

A natural consequence is that most such students are not too
interested in the kind of value addition that they get from such
places and are more focussed on the "hereafter". At the other end
of the chain, the companies that hire students are primarily
looking for "smarts" and "brights"; hence, the performance of the
student in a national level competitive examination (like JEE or
CAT) is far more important to them than their grades. The latter,
after all, are primarily an indication of how well they have
acquired knowledge which the companies are (mostly) not going to
make use of!

All of this is to argue that technical institutes in India
(which includes IISERs since we are called CFTIs --- centrally
funded technical institutes) should (IMNSHO) have more courses
that attempt to impart skills. Primarily, this means that we
should be oriented towards "problem solving" rather than
"information dissemination". Older readers will recognise that
this is an old mantra; the Deity/Devil is in the details of how
we implement it.

]]>Sat, 14 Feb 2015 05:57 GMTFrying Pan to Fireeducation/frying-pan-to-fire-2014-08-24-17-46http://www.imsc.res.in/~kapil/blog/education/frying-pan-to-fire-2014-08-24-17-46.html
A lot of people in academia cheered when Mr. Kapil Sibal was
re-shuffled out of the position of Minister for Human Resource.
Even those who agreed with his motives felt that his style of
functioning (especially on the issue of modifying the IIT-JEE)
was too autocratic. A similar cheer went through some academics
when Prof. Dinesh Singh was forced to withdraw the four-year
undergraduate programme at Delhi University; its introduction too
was "from above".

These cheers have probably died down, now that people have
realised that the current Minister for Human Resource, Ms. Smriti
Irani, like Mr. Arjun Singh before her, is not only autocratic
but has her target firmly pinned on the "numbers game".

We may bemoan the high-handed and aristrocratic people who
prefer to avoid the "cattle class". However, those of them who
have had a proper education know its value. According to the
others, educational innovation should only focus on how soon we
can produce (each year) 6000 people with PhD certificates and
proportional representation from all the reserved categories;
silimarly scaled up numbers of M.Tech's, MBA's, B.Tech's and so
on. This is because, for them, academia has no other purpose than
to "certify" the prepared-ness of various people for the national
and international job market. This is why "social justice", in
the eyes of the powerful, has never been about the proportional
right to quality education---it has been about a right to
proportional distribution of B.Tech. certificates from big-name
institutes.

The notion of "Institutional Autonomy" has, in any case, been
ground into the dust over the last many decades, so it is time
for academia to learn to appreciate a dictator who appreciates
academics, over one who does not.

]]>Sun, 24 Aug 2014 12:16 GMTResponse to a Questionnaire of the NCERTeducation/ncf-2014-06-01-20-03http://www.imsc.res.in/~kapil/blog/education/ncf-2014-06-01-20-03.html
I responded to a questionnare sent by the NCERT regarding the
teaching of mathematics and thought it would be worthwhile to
share the questions and answers.

Role and purpose of mathematics education: NCF-2005 has
envisioned that the main goal of mathematics education in
schools is the mathematisation of the child’s thinking.
Clarity of thought and pursuing assumptions to logical
conclusions is central to the mathematical enterprise. There
are many ways of thinking, and the kind of thinking one
learns in mathematics is an ability to handle abstractions,
and an approach to problem solving. Please comment on the
level of importance you would attach to different stages of
mathematics education and your views regarding its role and
purpose(s).

Mathematical learning/teaching must deal with three
components:

Development of the ability to carry out computations
(often without worrying about the context). This is sometimes
called Algebra.

Pinpointing one's doubts and analysing them through
logical thinking. This is sometimes called Analysis.

Constructing new mathematical objects out of old ones and
justifying these constructions. In school this is primarily
in the context of Geometry but later on it occurs elsewhere
as well.

All three aspects --- Algebra, Analysis and Geometry --- are
essential to Mathematics. One neglects one or the other at one's
own peril!

Concerns regarding mathematics: NCF-2005 has identified
the following core areas of concern: (a) A sense of fear and
failure regarding mathematics among a majority of children,
(b) A curriculum that disappoints both a talented minority as
well as the non-participating majority at the same time, (c)
Crude methods of assessment that encourage perception of
mathematics as mechanical computation, and (d) Lack of
teacher preparation and support in the teaching of
mathematics. (e) The emphasis on procedural skills rather
than on the understanding of mathematics (f) There are
concerns about the type and quality of the mathematics
education that students experience in schools. Systemic
problems further aggravate the situation, in the sense that
structures of social discrimination get reflected in
mathematics education as well. Especially worth mentioning in
this regard is the gender dimension, leading to a stereotype
that boys are better at mathematics than girls. Keeping these
concerns NCF-2005 is in action since last 6-7 years. We would
welcome your views on these issues or other concerns that you
may wish to raise. Also comments on the practical
experiences/ concerns on achievement via NCF-2005
recommendations are invited.

The point (d) about lack of trained teachers is the most
important concern. The primary difficulty is not the lack of
Mathematical training on the part of the teacher, though that is
certainly lacking in many cases. Rather, the problem is that the
teacher fails to see the classroom as one where she/he too can
learn. Thus, the teacher (and hence the student) fail to learn to
learn.

The second most important point is the focus on assessment (c)
which is seen as a punishment and reward system rather than as a
feedback mechanism by which the teacher and student figure out
what needs to be studied next.

The remaining problems stem from these two sources.

Recent developments in mathematics education: In the past
years, a revised mathematics curriculum has been implemented
at different stages of schooling. NCF-2005 has recommended:
(a) Shifting the focus of mathematics education from
achieving ‘narrow’ goals to ‘higher’ goals, (b) Engaging
every student with a sense of success, while at the same time
offering conceptual challenges to the emerging mathematician,
(c) Changing modes of assessment to examine students’
mathematization abilities rather than procedural knowledge,
and (d) Enriching teachers with a variety of mathematical
resources. The shift in focus NCF-2005 proposes is from
mathematical content to mathematical learning environments,
where a whole range of processes take precedence: formal
problem solving, use of heuristics, estimation and
approximation, optimisation, use of patterns, visualisation,
representation, reasoning and proof, making connections,
mathematical communication. Please comment on the impact of
these changes and whether they go far enough to address the
problems in mathematics that have been identified. Also
comments on the practical experiences/ concerns on
achievement via NCF-2005 recommendations are invited.

As mentioned above, no amount of twisting and turning with the
curriculum and contents will have a significant impact unless
such changes can help motivate the teachers better.

Also mentioned above is the need to retain the computational
skill component of mathematics. One learns to ride a cycle before
one learns how it is made and one learns to nurture and to grow
plants in a garden well before one learns the chemical and
biological processes behind agriculture. Of course, mathematics
is distinct in that the process of computation can be analysed
and re-built from the ground up. This does not mean that one
should start with such a (de-)(re-)construction.

A bad teacher can teach computational skills (through hateful
drill) which still will be useful, just as a bad programmer can
still write ugly computer programs which work. However, one needs
a good teacher if one is going to teach conceptualisation. While
the suggested aspects like visualisation are important, one
should ensure that one does not throw out the baby with the bath
water.

Current trends in mathematics education: A crucial
implication of recommended shift lies in offering a
multiplicity of approaches, procedures, solutions. NCF-2005
see this as crucial for liberating school mathematics from
the tyranny of the one right answer, found by applying the
one algorithm taught. Such learning environments invite
participation, engage children, and offer a sense of success.
Please comment on the relative merits / concerns of
applicability of such approaches in Mathematics classes at
different stages.

The algorithms that one learns have been developed through the
ages. This does not make them sacrosanct, but it does make it
important to learn to carry them out. For learning is, in good
part, the gathering of one's ancestral wisdom. At a certain age,
(usually around the time one enters high school) one learns to
question this wisdom --- and that should be encouraged, but one
can only do that if one has it on hand.

In mathematics, there is almost always only one "right"
answer. However, the emphasis should be on learning from the
process by which one obtained the answer, rather than reward for
the right answer and punishment for the wrong one.

It would be wrong to give students a sense of ambiguity about
the correctness of mathematics. An important aspect of
Mathematics (to all users of Mathematics) is its sense of
universal correctness. This becomes more nuanced as one studies
it deeper, but there is no doubt (amongst its practitioners) that
this is its primary purpose.

Teaching of Mathematics:- A VISION STATEMENT: NCF-2005
envisioned that, school mathematics takes place in a
situation where: • Children learn to enjoy mathematics: •
Children learn important mathematics: • Children see
mathematics as something to talk about, to communicate, to
discuss among themselves, to work together on. • Children
pose and solve meaningful problems: In school, mathematics is
the domain, which formally addresses problem solving as a
skill. • Children use abstractions to perceive relationships,
to see structure, to reason about things, to argue the truth
or falsity of statements. • Children understand the basic
structure of mathematics: Arithmetic, algebra, geometry and
trigonometry etc. • Teachers expect to engage every child in
class. Please comment on these visionary statements and on
the relationship between learning mathematics and these
processes. Please comment on the relative merits / concerns
of applicability of such approaches in Mathematics classes at
different stages. We would also welcome your views on the
students' learning intake and any suggestions you might have
for improvement/ updating.

The kind of child-centred learning is difficult if there are a
large number of students in each class. However, if the teacher
involves students in teaching each other, a lot is possible. At
the same time, the teacher needs to move from the role of giver
(or imparter) and be more involved in being trained
herself/himself. Adopting this role will help students see a live
example of continuous learning and thus pick up the habit
themselves Such a role will be difficult, if not impossible, in a
traditional-minded society such as ours.

Influence of the assessment: In terms of assessment,
NCF-2005 recommends that Board examinations be restructured,
so that the minimum eligibility for a State certificate be
numeracy, reducing the instance of failure in mathematics. On
the other hand, at the higher end, it recommends that
examinations be more challenging, evaluating conceptual
understanding and competence. Please give us your views on
the assessment of mathematics. Also, please comment on the
relative merits / concerns of applicability of such
approaches in Mathematics classes at different stages.

One of the greatest ills that plagues our education system is
that it is largely geared towards certification and eligibility.
Given societal needs this aspect of education may be unavoidable
and perhaps even necessary evil. We need to think about
mechanisms to give the students something more.

The primary purpose of in-classroom evaluation is as a
feedback mechanism that helps the student and the teacher improve
themselves and move forward. Focussing on grades at this stage is
definitely counter-productive. I believe that using this
assessment as part of the final grade has reduced its utility for
this reason. Thus internal evaluation should remain internal!

Coaching/drilling for certification examinations may need to
be separated from this process of classroom learning. This is
already happening through coaching classes. While it may be true
that coaching only helps students pass the "test of fire" (and
then feel "burnt out"), the drill and stamina development is not
unimportant if carried out in moderation. In today's society, one
does need to learn how to give examinations, appear for
interviews etcetera.

Syllabus levels and Curricular Choices: When it comes to
curricular choices, NCF-2005 recommends moving away from the
structure of tall and spindly education (where one concept
builds on another, culminating in university mathematics), to
a broader and well-rounded structure, with many topics
“closer to the ground”. If accommodating processes like
geometric visualisation can only be done by reducing content,
NCF-2005 suggests that content be reduced rather than
compromise on the former. Moreover, it suggests a principle
of postponement: in general, if a theme can be offered with
better motivation and applications at a later stage, wait for
introducing it at that stage, rather than go for technical
preparation without due motivation. As a practitioner, how
you feel about these recommendations in current syllabus and
its practices. Also, please comment on the issues, if any and
on how they might be addressed within the current
review.

At the shallow end of a pool one can learn not to fear water,
and one can learn how to push it around to get an idea of the
"theory" of swimming. However, one cannot learn swimming by
wallowing in shallow waters!

Numerous examples of the above kind have already been provided
before to say that there does not seem to be a reason to delay
the development of skills until one has learned the concepts
behind their operation. This does not mean that one should
neglect analysis and construction. However, it is acceptable for
the latter two to lag behind the pace at which one develops
skills. Just like singing, dancing and playing football, it is
easier to pick up a facility with numbers and symbols at an early
age. It is only a few highly motivated individuals who show the
courage and dedication required to learn these skills when they
are beyond their adolescent years.

Student achievement in mathematics: In previous questions
considers the evidences of learning Mathematics. How
effective, in your view, would each of the following measures
be in improving the performance of students in mathematics
assessment?

Remark: In question (i) below, this additional class time
should not be used to introduce more material!

(i) allocation of more class time to mathematics

effective

(ii) better pre-service and inservice education for teachers of mathematics

very effective

(iii) improved mathematics textbooks and other learning resources

effective

(iv) provision of learning support for students who are experiencing difficulties with the subject

very effective

(v) provision of ‘general’ as well as ‘specialist’ mathematics courses

effective

(vi) increased emphasis in examination questions on the application of mathematics to real-world problems

not effective

(vii) the introduction of additional forms of assessment, such as coursework

not effective

(viii) improving the perception of mathematics among parents and the general public

very effective

Teaching and learning in mathematics: What do you observe
that currently Mathematics classrooms indicates that
mathematics is taught and learned in a ‘traditional’ manner,
mainly involving teacher exposition or demonstration of
procedural skills and techniques for answering
examination-type questions, followed by student practice of
these techniques (in class or as homework) using similar
questions. There appears to be little or no emphasis on
students understanding the mathematics involved, or on its
application in different or unfamiliar contexts. Please
comment on the strengths and weaknesses of this approach. We
would also welcome your views on change in teaching and
learning approach in Mathematics, the degree to which
syllabus change, assessment change, teacher professional
development and support would contribute to bringing about
changes in teaching and learning.

The current mathematics classrooms are limited, but what they
are teaching is not unrelated to mathematical education. Thus the
current approach needs to be supplemented with additional work on
discussing the concepts, analysing doubts and attempting to
construct new concepts. For this to work, the teacher must be
also be encouraged to learn in the classroom along with her/his
students. The teacher can learn to teach better and also learn to
analyse and evaluate any mathematical ideas that are
discussed.

A certain amount of change in the syllabus (say about 10%) is
required to "keep up with the times". Even though Mathematics is
eternal, tastes and utility of mathematical ideas changes with
time.

In terms of assessment, it seems to me that involving internal
assessment in the certification process (of big competitive
examinations) is a failed experiment. It would be far better if
the former is kept separate from the latter so that classroom
assessment can serve its true goal of self-improvement.

Attitudes to and beliefs about mathematics: NCF-2005 – and
research papers on international trends in mathematics
education – raises, on a number of occasions, issues
surrounding the perceptions, attitudes and beliefs that exist
in relation to mathematics, such as • the view that
mathematics is a difficult subject • negative attitudes
towards mathematics including, for some, a ‘fear’ of the
subject • the perception and advocacy of mathematics,
particularly Higher level mathematics, as an elite subject
for only the ‘best’ students • research findings that suggest
a connection between teachers’ views of mathematics and their
approach to teaching it. We would welcome your views on these
or other issues associated with mathematics.

Since I have never feared mathematics myself, it is difficult
for me to pronounce judgement on why someone may fear or hate it.
However, I did fear and hate Biology and History in school since
these subjects seemed to rely excessively on memorisation and
elaborate descriptions, in place of analysis and accurate
summaries. I have since come to realise that my vision of these
subjects was myopic. I cannot with certainty say that it was due
to bad teaching either!

Perhaps it is pessimistic to say so, but it seems there will
always be some students who will fear and hate mathematics (or
any subject) just as there will be some who love it. This will be
independent of the quality of their teachers, the curriculum, the
contents or the books.

Just as it would be wrong to focus on the best students, it is
wrong to focus on the weak ("no child left behind").

Our primary job is to ensure that the large (97%) that sits
around the middle of the class does get a good mathematical
education---algebra, analysis and geometry/construction in
roughly equal parts.

Other influences: The discussion paper draws attention to
a range of other cross-cutting themes or issues that affect
mathematics education in schools: • cultural issues related
to the value of education in general and mathematics
education in particular • equality issues (gender, uptake and
achievement; socio-economic factors; educational
disadvantage; students with disabilities or special
educational needs) • recent developments in, and availability
of, information and communications technology (ICT) in
schools. Please comment on any of these issues, or on other
factors that impact on mathematics education in schools.

To the extent that computers provide us with (yet another)
example of the effectiveness of a mathematical approach, and to
the extent that they can help us handle computations that cannot
be done by hand, it would be good to involve them in mathematical
learning.

At the same time, it must be said that excessive exposure to
excellence and power can discourage. Just as it may be bad for a
youngster to watch too much IPL during the hours when she/he
should be playing cricket, it will hurt a young student of
mathematics to use a computer to carry out calculations which are
instructive to carry out "by hand".

Conclusion
The purpose of this review is to map out the direction that must
be taken in planning curriculum and assessment provision for
mathematics education at different school stages in the years
ahead while reviewing the NCF-2005. Please use the space below
to make any additional comments on current issues in mathematics
education or to give us your views regarding its future.

There is certainly scope for continuous improvement in the
teaching of Mathematics. Some mathematical skills which were
thought "advanced" 200 years ago could even be considered
"essential" for the generations to come.

Elsewhere, I have pronounced that mathematics is "conscious"
abstraction in the following sense. We make abstractions
unconsciously all the time. However, these are often very
individualised. In order to teach these abstractions to others we
must understand our own thought process and go through the steps
consciously. Only then can we turn these processes into theorems
and algorithms ... and thus into mathematics.

Academics and academia are themselves responsible for a lot of
issues.

General whining: Academics tend to whine a lot. This does
not (necessarily) mean that they are unhappy. A good part of
academic training is learning how to spot mistakes. Thus an
academic tends to find flaws all around her/him.

Teaching/learning: Research is not something you can teach
in a classroom. It has to be done. As a result, research
students are increasingly annoyed at their "teacher"'s
inability to teach. Most students never make the transition
to "learning" rather than "being taught".

Excluded Middle: While it is still true that the
"genius!"-types continue to be found in academia, the middle
tier of bright people are increasingly going elsewhere. This
is not sustainable in the long run. There must be a "We are
the 99%" movement for people to take control of their
science/knowledge --- but who will lead it?!

Exponential growth: Research requires sustained mental
growth. This is a steep curve and many feel like stepping
off. There are careers where one can solve problems from day
to day or week to week without worrying about becoming
obsolete. Academia is not one of them.

Specialisation: Getting a PhD can be described as
"becoming the worlds foremost expert on almost nothing". This
can also be described as digging a very deep well which is
only wide enough for one person (read under-nourished
graduate student).

As a result of one or more of these, someone who completes a
PhD thesis often feels disheartened. However, here are some
things to look forward to:

General whining: People in academia remember (and
embellish) their stories as a way of substantiating their
whining. Many of these stories are entertaining and almost all
are educational. This form of anecdotal learning about one's
workplace has no parallels in the startup culture of today.
Perhaps working for some of the dinosaurs like IBM, AT&T
will be similar --- even those can't compare with 400 year-old
oral histories.

Teaching/learning: There is no better place to learn a
subject than in a classroom --- as a teacher. More seriously,
preparing to teach a class or preparing exercises is one of the
ways to learn something really well. As someone once said, you
have not learned something properly until you have taught
it.

Excluded Middle: "A cat may look at a king." Being in
academics allows one to challenge and bring the lofty to earth.
A "genius" may (and often does) ignore those who are not
academics when they pose uncomfortable questions. A
middle-level academic is not so easy to dismiss.

Exponential Growth: There is tremendous opportunity for
doing "new stuff" in academia. As compared with any other
career, it is easiest to justify spending time on "non-core"
material in academics---like Alice, we have "to run very fast
to stay in the same place".

Specialisation: If training for a PhD can be seen as
training oneself to become a specialist, then there is no
reason one cannot iterate this and become a specialist in many
things. On the other hand, some others choose to "widen the
well and let other people in"!

There is no doubt that academia needs to break out of its
slumber, but who better than young, disgruntled PhD students to
do so?!

]]>Mon, 28 Apr 2014 05:48 GMTAcademics: Evolution or Revolutioneducation/academia-2013-09-15-11-11http://www.imsc.res.in/~kapil/blog/education/academia-2013-09-15-11-11.html
There are quite a few articles (for example, see here and
here) that are bitterly critical of academia and
research.

It is easy to dismiss these as cases of "burnout" or
"frustration". It is equally easy to dismiss these dismissals(!)
as "status-quo-ism" and "defense by the entrenched".

On the one hand, as a mathematician who does not depend on
extensive funding like some other sciences, or on an army of
graduate students to help me carry out my research, much of what
is described has not been personally experienced by me. That
said, there are issues raised in these articles --- like
the problem of evaluation of "merit" --- that cut across all
academia. Moreover, one has heard first-hand or second-hand
accounts of research in a number of sciences which seems to
mirror what is written --- about the "slavery" of PhD students,
for example.

Confession time: I enjoy the entire experience of academic
life --- right from the hours spent in the library cracking one's
head over a one or two pages, and the hours spent in the
laboratory tinkering with stuff to get it to work (of late this
was more often about debugging code on a computer), and the hours
spent discussing theory with colleagues and students, to being
part of campus life in various other ways.

So when I hear people describing their experience with
academics as being bitter, it is perplexing to say the least.

Are the problems that face academia in the nature of "bugs"
that can be fixed by "re-factoring" or are they serious enough
that one needs a "re-write"? Evolution or revolution? Most
articles and posts that describe the problems are like honey-pots
for those who would like to revolutionise or replace current
practices. Phrases like "paradigm shift", "holistic research" and
"relevance and utility" are bandied about.

On the other hand, senior academicians try to "explain away"
the problems by saying that academia should be thought of as an
industry and each university, institute or research centre as an
enterprise. Any enterprise has its own social, economic and
political strucures. The anarchic pursuit of knowledge, whether
for its own sake or for the common good of mankind, may seem to
be lost in personal or organisational goals that seem far more
crass and mundane. The "defenders of the faith" point out that
this no different from any other human activity that grows beyond
a certain size. Some may also say that compared with other large
human undertakings --- for example, the banking and finance
sector --- the academic community offers more room for
diversity.

It may very well be the case that the questions asked and
solved by "big (money) science" cannot be unraveled by small,
informal, essentially anarchic organisations. Can the lack of
sufficiently many motivated and competent teachers, that
currently plagues education, really be solved by MOOCs and "Khan
Academy" clones?

Questions like these are difficult and each one of us will
have to make choices about whether to "change the system from
within" or start on a radical new approach. The first often looks
like a compromise that will only strengthen existing institutions
along with their known flaws, while the latter runs the risk of
fading into irrelevance. Neither risk must prevent those who
belive in these approaches from trying to correct existing
wrongs.